This book is devoted to the structure of the absolute Galois groups
of certain algebraic extensions of the field of rational numbers.
Its main result, a theorem proved by the authors and Florian Pop in
2012, describes the absolute Galois group of distinguished
semi-local algebraic (and other) extensions of the rational numbers
as free products of the free profinite group on countably many
generators and local Galois groups. This is an instance of a
positive answer to the generalized inverse problem of Galois
theory. Adopting both an arithmetic and probabilistic approach, the
book carefully sets out the preliminary material needed to prove
the main theorem and its supporting results. In addition, it
includes a description of Melnikov's construction of free products
of profinite groups and, for the first time in book form, an
account of a generalization of the theory of free products of
profinite groups and their subgroups. The book will be of interest
to researchers in field arithmetic, Galois theory and profinite
groups.
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